SEISMIC DESIGN GUIDE FOR MASONRY BUILDINGS CHAPTER 1. Canadian Concrete Masonry Producers Association

Transcription

1 SEISMIC DESIGN GUIDE FOR MASONRY BUILDINGS CHAPTER 1 Donld Anderson Svetln Brzev Cndin Concrete Msonry Producers Assocition April 2009

2 DISCLAIMER While the uthors hve tried to be s ccurte s possible, they cnnot be held responsible for the designs of others tht might be bsed on the mteril presented in this document. The mteril included in this document is intended for the use of design professionls who re competent to evlute the significnce nd limittions of its contents nd recommendtions nd ble to ccept responsibility for its ppliction. The uthors, nd the Cndin Concrete Msonry Producers Assocition, disclim ny nd ll responsibility for the pplictions of the stted principles nd for the ccurcy of ny of the mteril included in the document. AUTHORS Don Anderson, Ph.D., P.Eng. Deprtment of Civil Engineering, University of British Columbi Vncouver, BC Svetln Brzev, Ph.D., P.Eng. Deprtment of Civil Engineering British Columbi Institute of Technology Burnby, BC TECHNICAL EDITORS Gry Sturgeon, P.Eng., Director of Technicl Services, CCMPA Bill McEwen, P.Eng., LEED AP, Eecutive Director, Msonry Institute of BC Dr. Mrk Hgel, EIT, Technicl Services Engineer, CCMPA GRAPHIC DESIGN Ntli Leposvic, M.Arch. COVER PAGE Photo credit: Bill McEwen, P.Eng. Grphic design: Mrjorie Greene, AICP COPYRIGHT Cndin Concrete Msonry Producers Assocition, 2009 Cndin Concrete Msonry Producers Assocition P.O. Bo 54503, 1771 Avenue Rod Toronto, ON M5M 4N5 Tel: (416) F: (416) Emil: informtion@ccmp.c Web site: The Cndin Concrete Msonry Producers Assocition (CCMPA) is non-profit ssocition whose mission is to support nd dvnce the common interests of its members in the mnufcture, mrketing, reserch, nd ppliction of concrete msonry products nd structures. It represents the interests of Region 6 of the Ntionl Concrete Msonry Assocition (NCMA).

3 Contents Summry Chpter 1 NBCC 2005 Seismic Provisions Objective: to provide bckground on seismic response of structures nd seismic nlysis methods nd eplin key NBCC 2005 seismic provisions of relevnce for msonry design DETAILED NBCC SEISMIC PROVISIONS Chpter 2 Seismic Design of Msonry Wlls to CSA S304.1 Objective: to provide bckground nd commentry for CSA S seismic design provisions relted to reinforced concrete msonry wlls, nd discuss the revisions in CSA S seismic design requirements with regrd to the 1994 edition DETAILED MASONRY DESIGN PROVISIONS Chpter 3 Summry of Chnges in NBCC 2005 nd CSA S Seismic Design Requirements for Msonry Buildings Objective: to provide summry of NBCC 2005 nd CSA S SUMMARY OF chnges with regrd to previous editions (NBCC 1995 nd CSA S NBCC AND 94) nd to present the results of design cse study of hypotheticl S304.1 low-rise msonry building to illustrte differences in seismic forces nd CHANGES msonry design requirements due to different site loctions nd different editions of NBCC nd CSA S304.1 Chpter 4 Design Emples Objective: to provide illustrtive design emples of seismic lod clcultion nd distribution of forces to members ccording to NBCC 2005, nd the seismic design of lodbering nd nonlodbering msonry elements ccording to CSA S DESIGN EXAMPLES Appendi A Appendi B Appendi C Appendi D Appendi E Comprison of NBCC 1995 nd NBCC 2005 Seismic Provisions Reserch Studies nd Code Bckground Relevnt to Msonry Design Relevnt Design Bckground Design Aids Nottion i

4 TABLE OF CONTENTS CHAPTER 1 1 SEISMIC DESIGN PROVISIONS OF THE NATIONAL BUILDING CODE OF CANADA Introduction Bckground Design nd Performnce Objectives Response of Structures to Erthqukes Elstic Response Inelstic Response Ductility A Primer on Modl Dynmic Anlysis Procedure Seismic Anlysis According to NBCC Seismic Hzrd Effect of Site Soil Conditions Methods of Anlysis Bse Sher Clcultions- Equivlent Sttic Anlysis Procedure Force Reduction Fctors R d nd R o Higher Mode Effects ( M v fctor) Verticl Distribution of Seismic Forces Overturning Moments ( J fctor) Torsion Configurtion Issues: Irregulrities nd Restrictions Deflections nd Drift Limits Dynmic Anlysis Method Soil-Structure Interction /1/

5 1 Seismic Design Provisions of the Ntionl Building Code of Cnd Introduction This chpter provides review of the seismic design provisions in the 2005 Ntionl Building Code of Cnd (NBCC 2005). Additionlly, there is n introduction to the dynmic nlysis of structures to ssist in understnding the NBCC provisions. Since there re mjor chnges to the seismic provisions reflected in NBCC 2005, some comprisons will be mde to the previous edition of the building code, NBCC 1995, nd this is covered in more detil in Appendi A. In the pst, building structures in mny res of Cnd did not hve to be designed for erthqukes. However, fter the NBCC 2005 ws issued nd dopted by the Provinces, structures in some dditionl res must now be designed for erthqukes, especilly if the structure is n importnt or post-disster building, or if it is locted on soft soil site. Since mny engineers in these regions hve not hd eperience in seismic design nd now my hve to include such design in their prctice, this guideline hs been prepred to eplin the seismic provisions included in the NBCC 2005 nd CSA S , nd to point out the recent chnges in these two documents s they pertin to msonry design. 1.2 Bckground Seismic design of msonry structures becme n issue following the 1933 Long Bech, Cliforni erthquke in which school buildings suffered dmge tht would hve been ftl to students hd the erthquke occurred during school hours. At tht time, seismic lterl lod equl to the product of seismic coefficient nd the structure weight hd to be considered in those res of Cliforni known to be seismiclly ctive. Strong motion instruments tht could mesure the pek ground ccelertion or displcement were developed round tht time, nd in fct, the first strong motion ccelerogrm ws recorded during the 1933 Long Bech erthquke. However, in this er the most widely used strong ground motion ccelertion record ws mesured t El Centro during the 1940 Imperil Vlley erthquke in southern Cliforni. The 1940 El Centro record becme fmous nd is still used by mny reserchers studying the effect of erthqukes on structures. With the vilbility of ground motion ccelertion records (lso known s ccelertion time history records), it ws possible to determine the response of simple structures modelled s single degree of freedom systems. After computers becme vilble in the 1960s it ws possible to develop more comple models for nlyzing the response of lrger structures. The dvent of computers hs lso hd huge impct on the bility to predict the ground motion hzrd t site, nd in prticulr, on probbilistic predictions of hzrd on which the NBCC seismic hzrd model is bsed. 4/1/

6 1.3 Design nd Performnce Objectives For mny yers, seismic design philosophy hs been founded on the understnding tht it would be too epensive to design most structures to remin elstic under the forces tht the erthquke ground motion cretes. Accordingly, most modern building codes llow structures to be designed for forces lower thn the elstic forces with the result tht such structures my be dmged in n erthquke, but they should not collpse, nd the occupnts should be ble to sfely evcute the building. The pst nd present NBCC editions follow this philosophy nd llow for lterl design forces smller thn the elstic forces, but impose detiling requirements so tht the inelstic response remins ductile nd brittle filure is prevented. Reserch studies hve shown tht for most structures, the lterl displcements or drifts re bout the sme irrespective of whether the structure remins elstic or it is llowed to yield nd eperience inelstic (plstic) deformtions. This is known s the equl displcement rule nd will be discussed lter in this chpter, s it forms the bsis for mny of the code provisions. The seismic response of building structure depends on severl fctors, such s the structurl system nd its dynmic chrcteristics, the building mterils nd design detils, but probbly the most importnt is the epected erthquke ground motion t the site. The epected ground motion, termed the seismic hzrd, cn be estimted using probbilistic methods, or be bsed on deterministic mens if there is n dequte history of lrge erthqukes on identifible fults in the immedite vicinity of the site. Cnd generlly uses probbilistic method to ssess the seismic hzrd, nd over the yers, the probbility hs been decresing, from roughly 40% chnce (probbility) of being eceeded in 50 yers in the 1970s (corresponding to 1/100 per nnum probbility, lso termed the 100 yer erthquke), to 10% in 50 yer probbility in the 1980s (the 475 yer erthquke), to finlly 2% in 50 yer probbility (the 2475 yer erthquke) used for NBCC The ltest chnge ws mde so tht the risk of building filure in estern nd western Cnd would be roughly the sme (Adms nd Atkinson, 2003), s well s to recognize tht n cceptble probbility of severe building dmge in North Americ from seismic ctivity is bout 2% in 50 yers. Despite the lrge chnges over the yers in the probbility level for the seismic hzrd determintion, the seismic design forces hve not chnged pprecibly becuse other fctors in the NBCC design equtions hve chnged to compenste for these higher hzrd vlues. Thus, while the code seismic design hzrd hs been rising over the yers, the seismic risk of filure of buildings designed ccording to the code hs not chnged gretly. A comprison of building designs performed ccording to the NBCC 1995 nd the NBCC 2005 will show n increse in design level forces in some res of Cnd nd decresed level in other res, however it is epected tht the overll difference between these designs is not significnt (see Appendi A for more detils). The NBCC 2005 hs tken more rtionl pproch towrds seismic design thn hve previous editions, in tht the seismic hzrd hs been ssessed for certin probbility relted to risk of severe building dmge, with the building designed with no empiricl or clibrting fctors. The rel strength of the building hs been utilized in the design, so tht t this level of ground motion it should not collpse but could be severely dmged. Thus, the probbility of severe dmge or ner collpse is bout 1/2475 per nnum, or bout 2% in the predicted 50- yer life spn of the structure. When compred to wind or snow lods, which re bsed on the 1 4/1/

7 in 50 yer probbility of not being eceeded, the 1 in 2475 yer probbility for seismic design ppers inconsistent. However, unlike design for seismic lods, design for wind nd snow lods uses lod nd mteril performnce fctors, nd so the resulting probbility of filure is epected to be smller thn tht for erthqukes. Seismic design does use mteril resistnce fctors, φ fctors, in ssessing member cpcity, but they re effectively cncelled out by the overstrength fctor, R (which will be described lter), used to reduce the seismic forces. o Work on new model codes round the world is leding to wht is described s, Performnce Bsed Design, concept tht is lredy being pplied by some designers working with owners who hve concerns tht building dmge will hve n dverse effect on their bility to mintin their business. NBCC 2005 only ddresses one performnce level, tht of collpse prevention nd life sfety, nd is essentilly mute on servicebility during smller seismic events tht re epected to occur more frequently. Performnce bsed design ttempts to minimize the cost of erthquke losses by weighing the cost of repir, nd cost of lost business, ginst n incresed cost of construction. 1.4 Response of Structures to Erthqukes Elstic Response When n erthquke strikes, the bse of building is subject to lterl motion while the upper prt of the structure initilly is t rest. The forces creted in the structure from the reltive displcement between the bse nd upper portion cuse the upper portion to ccelerte nd displce. At ech floor the lterl force required to ccelerte the floor mss is provided by the forces in the verticl members. The floor forces re inertil forces, not eternlly pplied forces such s wind lods, nd eist only s long s there is movement in the structure. Erthqukes cuse the ground to shke for reltively short time, 15 to 30 seconds of strong ground shking, lthough movements my go on for few minutes. The motion is cyclic nd the response of the structure cn only be determined by considering the dynmics of the problem. A few importnt dynmic concepts re discussed below. Consider simple single-storey building with msonry wlls nd flt roof. The building cn be represented by Single Degree of Freedom (SDOF) model (lso known s stick model) s shown in Figure 1-1. The mss, M, lumped t the top, represents the mss of the roof nd frction of the totl wll mss, while the column represents the combined wll stiffness, K, in the direction of erthquke ground motion. If n erthquke cuses lterl deflection, Δ, t the top of the building, Figure 1-1b, nd if the building response is elstic with stiffness, K, then the lterl inertil force, F, cting on the mss M will be F = K Δ When the mss of SDOF un-dmped structure is llowed to oscillte freely, the time for structure to complete one full cycle of oscilltion is clled the period, T, which for the SDOF system shown is given by M T = 2π (seconds) K Insted of period, the term nturl frequency, ω, is often used in seismic design. It is relted to the period s follows 4/1/

8 K ω = 2 π T = (rdins/sec) M Frequency is sometimes lso epressed in Hertz, or cycles per second, insted of rdins/sec, denoted by the symbol ω, where ω cps 1 = = T ω 2π cps Figure 1-1. SDOF system: ) stick model; b) displced position. As the structure vibrtes, there is lwys some energy loss which will cuse decrese in the mplitude of the motion over time - this phenomenon is clled dmping. The etent of dmping in building depends on the mterils of construction, its structurl system nd detiling, nd the presence of rchitecturl components such s prtitions, ceilings nd eterior wlls. Dmping is usully modelled s viscous dmping in elstic structures, nd hysteretic dmping in structures tht demonstrte inelstic response. In seismic design of buildings, dmping is usully epressed in terms of dmping rtio, β, which is described in terms of percentge of criticl viscous dmping. Criticl viscous dmping is defined s the level of dmping which brings displced system to rest in minimum time without oscilltion. Dmping less thn criticl, n under-dmped system, llows the system to oscillte; while n over-dmped system will not oscillte but tke longer thn the criticlly dmped system to come to rest. Dmping hs n influence on the period of vibrtion, T, however this influence is miniml for lightly dmped systems, nd in most cses is ignored for structurl systems. For building pplictions, the dmping rtio cn be s low s 2%, lthough 5% is used in most seismic clcultions. Dmping in structure increses with displcement mplitude since with incresing displcement more elements my crck or become slightly nonliner. For liner seismic nlysis viscous dmping is usully tken s 5% of criticl s the structurl response to erthqukes is usully close to or greter thn the yield displcement. A smller vlue of viscous dmping is usully used in nonliner nlyses s hysteretic dmping is lso considered. One of the most useful seismic design concepts is tht of the response spectrum. When structure, sy the SDOF model shown in Figure 1-1, is subjected to n erthquke ground motion, it cycles bck nd forth. At some point in time the displcement reltive to the ground nd the bsolute ccelertion of the mss rech mimum, Δ m nd m, respectively. Figure 1-2 shows the mimum displcement plotted ginst the period, T. Denote the period 4/1/

9 of this structure s T 1. If the dynmic properties, i.e. either the mss or stiffness chnge, the period will chnge, sy to T 2. As result, the mimum displcement will chnge when the structure is subjected to the sme erthquke ground motion, s indicted in Figure 1-2b. Repeting the bove process for mny different period vlues nd then connecting the points produces plot like tht shown in Figure 1-2c, which is termed the displcement response spectrum. The spectrum so determined corresponds to specific input erthquke motion nd specific dmping rtio, β. The sme type of plot could be constructed for the mimum ccelertion, m, rther thn the displcement, nd would be termed the ccelertion response spectrum. Figure 1-2. Development of displcement response spectrum - mimum displcement response for different periods T : ) T = T1 ; b) T = T2 ; c) mny vlues of T. Figure 1-3 shows the displcement response spectrum for the 1940 El Centro erthquke t different dmping levels. Note tht the displcements decrese with n increse in the dmping rtio, β, from 2% to 10%. Figure 1-3b shows the ccelertion response spectrum for the sme erthquke. For the smll mount of dmping present in the structures, the mimum ccelertion, m, occurs t bout the sme time s the mimum displcement, Δ m, nd these two prmeters cn be relted s follows 2 2π m = Δ m T Thus, by knowing the spectrl ccelertion, it is possible to clculte the displcement spectrl vlues nd vice vers. It is lso possible to generte response spectrum for mimum velocity. Ecept for very short nd very long periods, the velocity, v m, is closely pproimted by 2π vm = Δ m T This is generlly clled the pseudo velocity response spectrum s it is not the true velocity response spectrum. 4/1/

10 ) b) Figure 1-3. Response spectr for the 1940 El Centro NS erthquke t different dmping levels: ) displcement response spectrum; b) ccelertion response spectrum. The response spectrum cn be used to determine the mimum response of SDOF structure, given its fundmentl period nd dmping, to specific erthquke ccelertion record. Different erthqukes produce widely different spectr nd so it hs been the prctice to choose severl erthqukes (usully scled) nd use the resulting verge response spectrum s the design spectrum. For yers, the NBCC seismic provisions hve used this procedure where the design spectrum for site ws described by one or two prmeters, either pek ground ccelertion nd/or pek ground velocity, tht were determined using probbilistic mens. 4/1/

11 More recently, probbilistic methods hve been used to determine the spectrl vlues t site for different structurl periods. Figure 1-4 shows the 5% dmped ccelertion response spectrum for Vncouver used in developing the NBCC This is uniform hzrd response spectrum, i.e., spectrl ccelertions corresponding to different periods re bsed on the sme probbility of being eceeded, tht is, 2% in 50 yers. This will be discussed further in Section Figure 1-4. Uniform hzrd ccelertion response spectrum for Vncouver, 2% in 50 yer probbility, 5% dmping Inelstic Response For ny given erthquke ground motion nd SDOF elstic system it is possible to determine the mimum ccelertion nd the relted inerti force, F el, nd the mimum displcement, Δ el. Figure 1-5 shows force-displcement reltionship with the mimum elstic force nd displcement indicted. If the structure does not hve sufficient strength to resist the elstic force, F el, then it will yield t some lower level of inerti force, sy t lterl force level, F y. It hs been observed in mny studies tht structure with nonliner cyclic force-displcement response similr to tht shown in Figure 1-5b will hve mimum displcement tht is not much different from the mimum elstic displcement. This is indicted in Figure 1-5c where the inelstic (plstic) displcement, Δ u, is shown just slightly greter thn the elstic displcement, Δ el. This observtion hs led to the equl displcement rule, n empiricl rule which sttes tht the mimum displcement tht the structure reches in n erthquke is independent of its yield strength, i.e. irrespective of whether it demonstrtes elstic or inelstic response. The equl displcement rule is thought to hold becuse the nonliner response softens the structure nd so the period increses, thereby giving rise to incresed displcements. However, t the sme time, the yielding mteril dissiptes energy tht effectively increses the dmping (the energy dissiption is proportionl to the re enclosed by the force-displcement loops, termed hysteresis loops). Incresed dmping tends to decrese the displcements; therefore, it is possible tht the two effects blnce one nother with the result tht the elstic nd inelstic displcements re not significntly different. 4/1/

12 Figure 1-5. Force-displcement reltionship: ) elstic response; b) nonliner (inelstic) response; c) equl displcement rule. There re limits beyond which the equl displcement rule does not hold. In short period structures, the nonliner displcements re greter thn the elstic displcements, nd for very long period structures, the mimum displcement is equl to the ground displcement. However, the equl displcement rule is, in mny wys, the bsis for the seismic provisions in mny building codes which llow the structure to be designed for forces less thn the elstic forces. But there is lwys trde-off, nd the lower the yield strength, the lrger the nonliner or inelstic deformtions. This cn be inferred from Figure 1-5c where it is noted tht the difference between the nonliner displcement, Δ u, nd yield displcement, Δ y, which represents the inelstic deformtion, would increse s the yield strength decreses. Inelstic deformtions generlly relte to incresed dmge, nd the designer needs to ensure tht the strength does not deteriorte too rpidly with subsequent loding cycles, nd tht brittle filure is prevented. This cn be chieved by dditionl seismic detiling of the structurl members, which is usully prescribed by the mteril stndrds. For emple, in reinforced concrete structures, seismic detiling consists of dditionl confinement reinforcement tht ensures ductile performnce t criticl loctions in bems, columns, nd sher wlls. In reinforced msonry structures, it is difficult to provide similr confinement detiling, nd so restrictions re plced on limiting the reinforcement spcing, on levels of grouting, nd on certin strin limits in the msonry structurl components (e.g. sher wlls) which provide resistnce to seismic lods (see Chpter 2 for more detils on seismic design of msonry sher wlls) Ductility Ductility reltes to the cpcity of the structure to undergo inelstic displcements. For the SDOF structure, whose force-displcement reltion is shown in Figure 1-5c the displcement ductility rtio, μ, is mesure of dmge tht the structure might undergo nd cn be Δ epressed s μ Δ = Δ Δ u y The rtio between the mimum elstic force, reduction fctor, R, defined s F el, nd the yield force, F y, is given by the force 4/1/

13 R = F F el y If the mteril is elstic-perfectly plstic, i.e. there is no strin hrdening s it yields (see Figure 1-5b), nd if Δ is equl to Δ, then it cn be shown tht μ is equl to R. u el For different types of structures nd detiling requirements, most building codes tend to prescribe the R vlue while not mking reference to the displcement ductility rtio, μ, thus Δ implying tht the μ nd R vlues would be similr. Δ A Primer on Modl Dynmic Anlysis Procedure The min objective of this section is to eplin how more comple multi-degree-of-freedom structures respond to erthquke ground motions nd how such response cn be quntified in form useful for structurl design. This bckground should be helpful in understnding the NBCC seismic provisions Multi-degree-of-freedom systems The ide of modelling the building s SDOF structure ws introduced in Section 1.4.1, nd the dynmic response to erthquke ground motions ws developed in terms of response spectrum. Such simple model might well represent the lterl response of single storey wrehouse building with fleible wlls or brcing system, nd with rigid roof system where the roof comprises most of the weight (mss) of the structure. However, this is not good model for msonry wrehouse with metl deck roof, where the wlls re quite stiff nd the deck is fleible nd light reltive to the wlls. Such system requires more comple model using multi-degree-of-freedom (MDOF) system. A sher wll in multi-storey building is nother emple of MDOF system. Figure 1-6 shows two emples of MDOF structures. A simple four-storey structure is shown in Figure 1-6, nd simple MDOF model for this building consists of column representing the stiffness of verticl members (sher wlls or frmes), with the msses lumped t the floor levels. If the floors re rigid, it cn be ssumed tht the lterl displcements t every point in floor re equl, nd the structure cn be modelled with one degree of freedom (DOF) t ech floor level ( DOF cn be defined s lterl displcement in the direction in which the structure is being nlyzed). This will result in s mny degrees of freedom s number of floors, so this building cn be modelled s 4-DOF system. It must lso be ssumed tht there re no torsionl effects, tht is, there is no rottion of the floors bout verticl is (torsionl effects will be discussed lter in Section 1.5.9). The nlysis will be the sme irrespective of the lterl force resisting system ( sher wll or frme), side from detils in finding the lterl stiffness mtri for the floor displcements. The wrehouse building shown in Figure 1-6b is nother emple of MDOF structure. The wlls re treted s single column with some portion of the wll nd roof mss, M 1, locted t the top. The roof cn be treted s spring (or severl springs) with the remining roof mss, M 2, ttched to the spring(s). How much mss to ttch to ech degree of freedom, nd how to determine the stiffness of the roof, re mjor chllenges in this cse. Δ 4/1/

14 Figure 1-6. MDOF systems: ) multi-storey sher wll building; b) wrehouse with fleible roof Seismic nlysis methods The question of interest to structurl engineers is how to determine relistic seismic response for MDOF systems? The possible pproches re: sttic nlysis, nd dynmic nlysis (modl nlysis or time history method). The simplest method is the equivlent sttic nlysis procedure (lso known s the qusi-sttic method) in which set of sttic horizontl forces is pplied to the structure (similr to wind lod). These forces re ment to emulte the mimum effects in structure tht dynmic nlysis would predict. This procedure works well when pplied to smll, simple structures, nd lso to lrger structures if they re regulr in their lyout. NBCC 2005 specifies dynmic nlysis s the defult method. The simplest type of dynmic nlysis is the modl nlysis method. This method is restricted to liner systems, nd consists of dynmic nlysis to determine the mode shpes nd periods of the structure, nd then uses response spectrum to determine the response in ech mode. The response of ech 4/1/

15 mode is independent of the other modes, nd the modl responses cn then be combined to determine the totl structurl response. In the net section, the modl nlysis procedure will be eplined with n emple. The second type of dynmic nlysis is the time history method. This consists of dynmic nlysis model subjected to time-history record of n erthquke ground motion. Time history nlysis is powerful tool for nlyzing comple structures nd cn tke into ccount nonliner structurl response. This procedure is comple nd time-consuming to perform, nd s such, not wrrnted for low-rise nd regulr structures. It requires n dvnced level of knowledge of the dynmics of structures nd it is beyond the scope of this document. For detiled bckground on dynmic nlysis methods the reder is referred to Chopr (2007) Modl nlysis procedure: n emple Consider four-storey sher wll building emple such s tht shown in Figure 1-6. The building cn be modelled s stick model, with weight,w, of 2,000 kn lumped t ech floor level, nd uniform floor height of 3 m (see Figure 1-7). For simplicity, the wll stiffness nd the msses re ssumed uniform over the height. This model is MDOF system with four degrees of freedom consisting of lterl displcement t ech storey level. A MDOF system hs s mny modes of vibrtion s degrees of freedom. Ech mode hs its own chrcteristic shpe nd period of vibrtion. The periods re given in Tble 1-1, the four mode shpes re given in Tble 1-2 nd shown in Figure 1-7. In this emple, the stiffness hs been djusted to give first mode period of 0.4 seconds, which is representtive of four-storey structure bsed on simple rule-of-thumb tht the fundmentl period is on the order of 0.1 sec per floor. Note tht the first mode, lso known s the fundmentl mode, hs the longest period. The first mode is by fr the most importnt for determining lterl displcements nd interstorey drifts, but higher modes cn substntilly contribute to the forces in structures with longer periods. In this emple the mode shpes hve been normlized so tht the lrgest modl mplitude is unity. For liner elstic structures, the equtions governing the response of ech mode re independent of the others provided tht the dmping is prescribed in prticulr mnner. Thus, the response in ech mode cn be treted in mnner similr to SDOF system, nd this llows the mimum displcement, moment nd sher to be clculted for ech mode. In the finl picture, the modl responses hve to somehow be combined to find the design forces (this will be discussed lter in this section). Modl nlysis cn be performed by hnd clcultion for simple structure, however, in most cses, the use of dynmic nlysis computer progrm would be required. Knowing the mode shpes nd the mss t ech level, it is possible to clculte the modl mss for ech mode, which is given in Tble 1-1 s frction of the totl mss of the structure. The modl msses re representtive of how the mss is distributed to ech mode, nd the sum of the modl msses must dd up to the totl mss. When doing modl nlysis, sufficient number of modes should be considered so tht the sum of the modl msses dds up to t lest 90% of the totl mss. In the emple here this would indicte tht only the first two modes would need to be considered ( = 0.906). 4/1/

16 Figure 1-7. Four-storey sher wll building model nd modl shpes. As n emple of how the different modes cn be used to determine the structurl response, Figure 1-8 shows typicl design ccelertion response spectrum which will be used to determine the modl displcements nd ccelertions. The four modl periods re indicted on the spectrum (note tht only the first two periods re identified on the digrm; T 1 =0.40 nd T 2 =0.062 sec) nd the spectrl ccelertion S t ech of the periods is given in Tble 1-3. Figure 1-8. Design ccelertion response spectrum. A very useful feture of the modl nlysis procedure gives the bse sher in ech mode s product of the modl mss nd the spectrl ccelertion S for tht mode, s shown in Tble 1-3. For emple, the bse sher for the first mode is equl to (8000kN 0.696) 0.74 = 4127 kn). Note tht the spectrl ccelertion is higher for the higher modes, but becuse the modl mss for these modes is smller, the bse sher is smller. The inerti forces from ech floor mss ct in the sme directions s the mode shpe, tht is, some forces re positive while others re negtive for the higher modes (refer to mode shpes shown in Figure 1-7). It cn be seen from the figure tht the forces from the first mode ll ct in the sme direction t the sme time, while the higher modes will hve both positive nd negtive forces. Thus the bse sher from the first mode is usully lrger thn tht from the other modes. 4/1/

17 The modl bse shers shown in Tble 1-3 re the mimum bse shers for ech mode. It is very unlikely tht these forces will occur t the sme time during the ground shking, nd they could hve either positive or negtive signs. Summing the contribution of ech mode where ll vlues re tken s positive, known s the bsolute sum (ABS) method, produces very high upper bound estimte of the totl bse sher. Sttisticl nlyses hve shown tht the squreroot-of-the-sum-of-the squres (RSS) procedure, whereby the contribution of ech mode is squred, nd the squre root is then tken of the sum of the squres, gives resonbly good estimte of the modl sum, especilly if the modl periods re widely seprted. Tble 1-3 shows the bse sher vlues estimted by the two methods nd gives n indiction of the conservtism of the ABS method for this cse (totl bse sher of 6,462 kn), where the modl periods re widely seprted, nd use of the RSS method is pproprite since it gives lower totl bse sher vlue of 4,468 kn. Note tht there is third method tht is incorported in mny modl nlysis progrms clled the complete-qudrtic-combintion (CQC) method. This method should be used if the periods of some of the modes being combined re close together, s would be the cse in mny three-dimensionl structurl nlyses, but for most structures with well-seprted periods nd low dmping, the result of the RSS nd CQC methods will be nerly identicl (this is true for most two-dimensionl structurl nlyses). The mplitude of displcement in ech mode is dependent upon the spectrl ccelertion for tht mode nd its modl prticiption fctor, which is mesure of the degree to which certin mode prticiptes in the response. The vlue of the modl prticiption fctor depends on how the mode shpes re normlized, nd so will not be given here, however the vlues re smller for the higher modes with the result tht the displcements for the higher modes re generlly smller thn those of the first mode. The modl displcements re presented in Tble 1-4 (to three deciml plces, which is why some vlues re shown s zero) nd plotted in Figure 1-9 for the first two modes s well s the RSS vlue. In this emple, the influence of the two highest modes is very smll nd hs been omitted from the digrm. It is difficult to distinguish between the first mode displcements nd the RSS displcements in Figure 1-9; this is chrcteristic of structures with periods less thn bout 1 second, such s would be the cse for most msonry structures. Figure 1-9. Modl displcements. 4/1/

18 Modl nlysis gives the modl shers nd bending moments in ech member nd these vlues cn be used to generte the sher nd moment digrms. These re summrized in Tbles 1-5 nd 1-6, nd re grphiclly presented in Figure Only the results from the first two modes re shown s the higher modes contribute very little to the response. Ecept for some contribution to the shers, the second mode is insignificnt in contributing to the totl vlues clculted using the RSS method. ) b) Figure Modl nlysis results: ) sher forces; b) bending moments. The inerti force t ech floor for ech mode cn be determined by tking the difference between the sher force bove nd below the floor in question. Modl inerti forces long with the RSS vlues re summrized in Tble 1-7, nd show tht the higher modes t some levels contribute more thn the first mode. Note tht the sum of the inerti forces for ech mode is equl to the bse sher for tht mode. However, the sum of the RSS vlues of the floor forces t ech level is 6284 kn (obtined by dding vlues for storeys 1 to 4 in the lst column of the tble); this is not equl to the totl bse sher of 4468 kn found by tking the RSS of the bse shers in ech mode (see Tble 1-3). This demonstrtes the key rule in combining modl responses: only primry quntities from ech mode should be combined. For emple, if the designer is interested in the sher force digrm for the structure, it is necessry to find the sher forces in ech mode nd then combine these modl quntities using the RSS method. It is incorrect to find the totl floor forces t ech level from the RSS of individul modl vlues, nd then use these totl forces to drw the sher digrm. Even interstorey drift rtios, defined s the difference in the displcement from one floor to the net divided by the storey height, should be clculted for ech mode nd then combined using the RSS procedure. It would be incorrect to divide the totl floor displcements by the storey height; lthough in this emple since the deflection is lmost entirely given by the first mode this pproch would be very close to tht found using the RSS method. One of the disdvntges of modl nlysis is tht the signs of the forces re lost in the RSS procedure nd so equilibrium of the finl force system is not stisfied. Equilibrium is stisfied in ech mode, but this is lost in the procedure to combine modl quntities since ech quntity is squred. Tht is why it is importnt to determine quntities of interest by combining only the originl modl vlues. 4/1/

19 Comprison of sttic nd modl nlysis results The equivlent sttic force nlysis procedure, which will be presented in more detil in Section 1.5.4, hs been pplied to the four storey structure described bove for the spectrum shown in Figure 1-8. Tble 1-8 compres the results of the two types of nlyses. It cn be seen tht both the bse sher nd moment given by the modl nlysis method is bout 75% of tht given by the sttic method. This occurs with short period MDOF structures tht respond in essentilly the first mode becuse the modl mss of the first mode for wlls is bout 70 to 80% of the totl mss. The top displcement from the modl nlysis is 78% of the sttic displcement, nerly the sme s the rtio of the bse moments; this would be epected given tht the deflection is mostly tied to the moment. If the structure is single-storey, SDOF system, the two nlyses methods will give the sme result. But for MDOF systems, such s two-storey or higher buildings, dynmic nlysis will generlly result in smller forces nd displcements thn the sttic procedure. The floor forces from the two nlyses re quite different. The floor forces in the upper storeys obtined by modl nlysis re less thn the sttic forces, but in the lower storeys, reverse trend cn be observed. The reson for this is the contribution of the higher modes to the floor forces. It cn be seen in Tble 1-7, tht t the 2 nd storey, the second mode contribution is the lrgest of ll the modes. To ensure the required sfety level when seismic design is performed using the equivlent sttic nlysis procedure, NBCC 2005 seismic provisions (e.g. Cluse ) provides dditionl guidnce on the level of floor forces to be used in connecting the floors to the lterl lod resisting elements. 4/1/

20 Tble 1-1. Modl Periods nd Msses Mode Period (sec) Modl mss/ Totl mss Sum Tble 1-2. Mode Shpes Storey Mode Shpes 1 st mode 2 nd mode 3 rd mode 4 th mode Note: mode shpes re normlized to mimum of 1 Tble 1-3. Spectrl Accelertions, S, nd Bse Shers Mode Period (sec) Spectrl Accelertion S (g) Modl mss / Totl mss Bse Sher (kn) Totl bse sher ABS 6462 Totl bse sher RSS 4468 Note: totl weight = 8000 kn Tble 1-4. Modl Displcements Storey Modl Displcements (cm) 1 st mode 2 nd mode 3 rd mode 4 th mode RSS Bse /1/

21 Tble 1-5. Modl Sher Forces Storey Sher Forces (kn) 1 st mode 2 nd mode 3 rd mode 4 th mode RSS Tble 1-6. Modl Bending Moments Storey Bending Moments (knm) 1 st mode 2 nd mode 3 rd mode 4 th mode RSS Bse Tble 1-7. Modl Inerti Forces (Floor Forces) Storey Floor Forces (kn) 1 st mode 2 nd mode 3 rd mode 4 th mode RSS Sum Tble 1-8. Comprison of Sttic nd Dynmic Anlyses Results Storey Sher Forces (kn) Floor Forces (kn) Moments (knm) Deflections (cm) Sttic Modl (1) Sttic Modl (2) Sttic Modl (3) Sttic Modl (4) Bse Notes: (1) see Tble 1-5, lst column (2) see Tble 1-7, lst column; (3) see Tble 1-6, lst column; (4) see Tble 1-4, lst column. 4/1/

22 1.5 Seismic Anlysis According to NBCC 2005 This section presents nd eplins the relevnt seismic code provisions in NBCC Reference will be mde here to NBCC 1995 where pproprite, but Appendi A contins the pertinent 1995 code provisions nd comprison of the design forces from the two codes Seismic Hzrd (6) One of the mjor chnges to the seismic provisions between the 1995 nd 2005 editions of the NBCC is relted to the determintion of the seismic hzrd. The 1995 code ws bsed on probbilistic estimtes of the pek ground ccelertion nd pek ground velocity for probbility of eceednce of 1/475 per nnum (10% in 50 yers). For NBCC 2005, the seismic hzrd is bsed on 2% in 50 yers probbility (corresponding to 1/2475 per nnum), nd it is represented by the 5% dmped spectrl response ccelertion, S (T ). During the NBCC 2005 code development cycle, records becme vilble, nd the bility to compute how response spectrl vlues vry with mgnitude nd distnce from source to site gretly improved. Thus, it ws possible to compute probbilistic estimtes of spectrl ccelertion for different structurl periods, nd construct response spectrum where ech point on the spectrum hs the sme probbility of eceednce. Such spectrum is termed Uniform Hzrd Spectrum, or UHS. The ccelertion UHS for Montrel is shown in Figure Figure Uniform hzrd spectrum for Montrel (UHS), 2% in 50 yers probbility, 5% dmping. For design purposes, the NBCC 2005 does not use the UHS, but rther n pproimtion given by four period-spectrl vlues which re used to construct spectrum, S (T ), which is used s the bsis for the design spectrum. For mny loctions in the country, these vlues re specified in Tble C-2, Appendi C to the NBCC 2005, long with the pek ground ccelertion (PGA) for ech loction, which is used minly for geotechnicl purposes. For other Cndin loctions, it is possible to find the vlues online t: by entering the coordintes (ltitude nd longitude) of the loction. The progrm does not directly clculte the S (T ) vlues, but insted, interpoltes them from the known vlues t 4/1/

23 severl surrounding loctions. For detiled informtion on the models used s the bsis for the NBCC 2005 seismic hzrd provisions, the reder is referred to Adms nd Hlchuk (2003). Figure 1-12 shows the S (T ) spectrum for Montrel nd the corresponding UHS. Since S (T ) is constructed using only four points (corresponding to different periods), it is n pproimtion to the UHS, nd it lso reflects some conservtism in the code. At very short periods S (T ) is tken to be constnt t the S (0.2) vlue, nd it does not decrese to the PGA, which is the UHS vlue t zero period. This my pper to be very conservtive, but only few structures hve periods less thn 0.2 sec, nd there re other resons relted to the inelstic response of such short-period structures, to be conservtive in this region. Note tht mny low-rise msonry buildings my hve fundmentl period on the order of 0.2 sec. The dt needed to clculte the UHS vlues for lrge periods (over 2 seconds) is not vilble for ll regions in Cnd, nd so between 2 seconds nd 4 seconds, S (T ) is ssumed to vry s1 / T. Beyond 4 seconds there is even less dt, nd S (T ) is ssumed to be constnt t the S (4) vlue for periods lrger thn 4 seconds. S (T ) is defined s the design hzrd spectrum for sites locted on wht is termed soft rock or very dense soil. For sites situted on either hrder rock or softer soil the hzrd spectrum needs to be modified s discussed below. Figure S (T ) nd UHS spectrum for Montrel Effect of Site Soil Conditions In the NBCC 2005, the seismic hzrd given by the S (T ) spectrum hs been developed for site tht consists of either very dense soil or soft rock (Site clss C within NBCC 2005). If the structure is to be locted on soil tht is softer thn this, the ground motion my be mplified, or in the cse of rock or hrd rock sites, the motion will be de-mplified. In NBCC 2005 two site coefficients re provided to be pplied to the S (T ) spectrum to ccount for these locl ground conditions. The coefficients depend on the building period, level of seismic hzrd, s well s on the site properties, which re described in terms of site clsses. The NBCC 2005 site coefficients re more detiled thn the foundtion fctor, F, provided in previous code editions, but should better represent the effect of the locl soil conditions on the seismic response. 4/1/

24 Tble 1-9 ecerpted from NBCC 2005, describes si site clsses, lbelled from A to E, which correspond to different soil profiles (note tht the seventh clss, F, is one tht fits none of the first si nd would require specil investigtion). The site clsses re bsed on the properties of the soil or rock in the top 30 m. Site clss C is the bse clss for which the site coefficients re unity, i.e. it is the type of soil on which the dt used to generte the S ( T ) spectrum is bsed. The tble identifies three soil properties tht cn be used to identify the site clss; the best one being the verge sher wve velocity, V s, which is prmeter tht directly ffects the dynmic response. The site clss determintion is bsed on the weighted verge, of the property being considered, in the top 30 m, which for V s would correspond to the verge velocity it would tke for sher wve to trverse the 30 m depth. NBCC 2005 nd Commentry J (NRC, 2006) do not discuss the level from which the 30 m should be mesured. For buildings on shllow foundtions, the 30 m should be mesured from the bottom of the foundtion. However, if the building hs very deep foundtion where the ground motion forces trnsferred to the building my come from both friction t the bse nd soil pressures on the sides, the nswer is not so cler nd my require site specific investigtion to determine the ccelertions of the building foundtion. Tble 1-9. NBCC 2005 Site Clssifiction for Seismic Response (NBCC 2005 Tble A) Averge Properties in Top 30 m, s per Appendi A Site Clss Ground Profile Nme Averge Sher Wve Velocity, V s (m/s) Averge Stndrd Penetrtion Soil Undrined Sher Strength, s u Resistnce, N 60 A Hrd rock V s > 1500 Not pplicble Not pplicble B Rock 760 < V s 1500 Not pplicble Not pplicble C Very dense soil nd soft rock 360 < V s < 760 N 60 > 50 s u > 100kP D Stiff soil 180 < V s < < N 60 < < s u 100kP V s <180 N 60 < 15 s u < 50kP E Soft soil Any profile with more thn 3 m of soil with the following chrcteristics: plsticity inde: PI > 20 moisture content: w 40%; nd undrined sher strength: s u < 25 kp F Other soils (1) Site-specific evlution required Reproduced with the permission of the Ntionl Reserch Council of Cnd, copyright holder Notes: (1) Other soils include: ) liquefible soils, quick nd highly sensitive clys, collpsible wekly cemented soils, nd other soils susceptible to filure or collpse under seismic loding, b) pet nd/or highly orgnic clys greter thn 3 m in thickness, c) highly plstic clys (PI>75) more thn 8 m thick, d) soft to medium stiff clys more thn 30 m thick. The effect of the site clss on the response spectrum is given by the following two site coefficients: F, which modifies the spectrum S ( T ) in the short period rnge (see Tble 1-10), nd S T in the longer period rnge (see Tble 1-11). F, which modifies ( ) v 4/1/

25 Tble Vlues of F s Function of Site Clss nd S (0.2) (NBCC 2005 Tble B) Site Vlues of F Clss S (0.2) 0.25 S (0.2) = 0.50 S (0.2) = 0.75 S (0.2) =1.00 S (0.2) = 1.25 A B C D E F (1) (1) (1) (1) (1) Reproduced with the permission of the Ntionl Reserch Council of Cnd, copyright holder Notes: (1) See Sentence (5). Tble Vlues of F s Function of Site Clss nd S (1.0) (NBCC 2005 Tble C) v Site Vlues of F v Clss S (1.0) 0.1 S (1.0) = 0.2 S (1.0) = 0.3 S (1.0) =0.4 S (1.0) > 0.5 A B C D E F (1) (1) (1) (1) (1) Reproduced with the permission of the Ntionl Reserch Council of Cnd, copyright holder Notes: (1) See Sentence (5). Note tht the F nd F v vlues depend on the level of seismic hzrd s well s on the site soil clss. For soft soil sites (site clsses D nd E), motion from high hzrd event would led to higher sher strins in the soil, which gives rise to higher soil dmping nd reduced surfce motion, when compred to low hzrd motion. The softer the soil, s given by higher site clssifiction, the higher the site coefficients, ecept for few F vlues t high hzrd level. For rock nd hrd rock, the site coefficients will generlly be less thn unity. The F nd F fctors re pplied to the S ( T ) spectrum to give S ( T ) v spectrl ccelertion for the site. The clcultion of ( T ) emple., which is the design S vlues will be illustrted with n Figure 1-13 shows the design seismic hzrd spectrum, S (T ), for Vncouver for firm ground site, Clss C, nd soft soil site, Clss E. For Vncouver (Grnville nd 41 Ave): S ( 0.2) =0.96g, S ( 0.5) =0.66g, S ( 1.0) =0.34g, nd S ( 2.0) =0.17 (see Appendi C, NBCC 2005; note tht these vlues were tken from n erlier version of Tble C-2 nd re slightly different from the published vlues). Interpolting from the vlues in Tble 1-10 for site Clss E nd ( 0.2) nd from Tble 1-11 for S ( 1.0) =0.34g, gives F v =1.82. S =0.96g, gives F =0.932, The clcultions to determine S (T ) for the Clss E site re (see Cluse (6)): 4/1/

26 For T=0.2 sec: S(0.2) = F S (0.2) = =0.89 S(0.2)=0.89 For T=0.5 sec: S(0.5) = F v S (0.5) = = 1.2, or S(0.5) = F S (0.2) = = 0.89, whichever is smller Since the smller vlue governs, S(0.5)=0.89 For T=1 sec: S(1.0) = F v S (1.0) = = 0.62 S(1.0)=0.62 For T=2 sec: S(2.0) = F v S (2.0) = = 0.31 S(2.0)=0.31 For T 4 sec: S(T) = F v S (2.0)/2 = /2 = S(T 4.0)=0.155 The resulting S (T ) soil Clss C nd E design spectr for Vncouver re plotted in Figure Figure S(T) design spectr for Vncouver for site Clsses C nd E Methods of Anlysis NBCC 2005 prescribes two methods of clculting the design bse sher of structure. The dynmic method is the defult method, but the equivlent sttic method cn be used if the structure meets ny of the following criteri: () is locted in region of low seismic ctivity where I E F S ( 0.2) < ( I E is the erthquke importnce fctor of the structure s defined in Cluse (1)), (b) is regulr structure less thn 60 m in height with period, T, less thn 2 seconds in either direction ( T is defined s the fundmentl lterl period of vibrtion of the structure in the direction under considertion, s defined in Cluse (3)),or (c) is n irregulr structure, but does not hve Type 7 irregulrity, nd is less thn 20 m in height with period, T, less thn 0.5 seconds in either direction (see Section for more detils on irregulrities). The equivlent sttic method will be described in this section becuse it likely cn be used on the mjority of msonry buildings given the bove criteri, nd notwithstnding, if the dynmic method is used, it must be clibrted bck to the bse sher determined from the equivlent sttic nlysis procedure. Bsic concepts of the modl dynmic nlysis method were presented in Section 1.4.4, nd further discussion is offered in Section /1/